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Proc. R. Soc. A (2010) 466, 383–406doi:10.1098/rspa.2009.0394

Published online 21 October 2009

An analytical model for dispersion of pollutantsfrom a continuous source in the atmospheric

boundary layerBY PRAMOD KUMAR AND MAITHILI SHARAN*

Centre for Atmospheric Sciences, Indian Institute of Technology Delhi,Hauz Khas, New Delhi 110016, India

For the dispersion of a pollutant released from a continuous source in the atmosphericboundary layer (ABL), a generalized analytical model describing the crosswind-integratedconcentrations is presented. An analytical scheme is described to solve the resulting two-dimensional steady-state advection–diffusion equation for horizontal wind speed as ageneralized function of vertical height above the ground and eddy diffusivity as a functionof both downwind distance from the source and vertical height. Special cases of thismodel are deduced and an extensive analysis is carried out to compare the model withthe known analytical models by taking the particular forms of wind speed and verticaleddy diffusivity.

The proposed model is evaluated with the observations obtained from Copenhagendiffusion experiments in unstable conditions and Hanford and Prairie Grass experimentsin stable conditions. In evaluation of the model, a recently proposed formulation for thewind speed in the entire ABL is used. It is concluded that the present model is performingwell with the observations and can be used to predict the short-range dispersion froma continuous release. Further, it is shown that the accurate parameterizations of windspeed and eddy diffusivity provide a significant improvement in the agreement betweencomputed and observed concentrations.

Keywords: analytical model; pollutant dispersion; crosswind-integrated concentration;atmospheric boundary layer; variable wind speed and eddy diffusivity

1. Introduction

Dispersion of a continuous release of air pollutants in the atmosphericboundary layer (ABL) is an important problem owing to the complex natureof inhomogeneous sheared turbulence near a rough boundary, stability andmeteorological conditions such as wind, temperature inversion and foggyatmosphere. When a plume is dispersed in the ABL, its shape, evolution andinternal structure are determined by the interaction between the plume and the

*Author for correspondence ([email protected]).

Electronic supplementary material is available at http://dx.doi.org/10.1098/rspa.2009.0394 or viahttp://rspa.royalsocietypublishing.org.

Received 25 July 2009Accepted 16 September 2009 This journal is © 2009 The Royal Society383

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384 P. Kumar and M. Sharan


turbulent eddies that characterize the atmospheric motion. In spite of the recentadvances in the theory and the direct numerical simulation of turbulent dispersionin the ABL, there remains a need of a fast flexible way to predict the transportof pollutants and other substances derived directly or indirectly from distributedsurface sources in the atmosphere (Smith 2003).

The advection–diffusion equation (Seinfeld 1986) has long been used to describethe dispersion of air pollutants in the turbulent atmosphere. The use of analyticalsolutions of this equation was the first and remains the convenient way formodelling air pollution (Demuth 1978). The conventional air pollution modelassumes wind speed and eddy diffusivity as constant throughout the entire ABLin deriving the solution of the advection–diffusion equation. In the solutionobtained, the eddy diffusivity is parameterized in terms of dispersion parametersand downwind distance in analogy with the Gaussian distribution to describethe concentration close to the observations (Csanady 1973; Seinfeld 1986; Arya1999). In stationary and horizontally homogeneous conditions, observational andtheoretical studies show that the wind speed and eddy diffusivity vary withthe vertical height above the ground (Stull 1988). Based on Taylor’s (1921)analysis and statistical theory, it is argued (Arya 1995; Demael & Carissimo2008) that the eddy diffusivity depends on the downwind distance from thesource (Mooney & Wilson 1993). Thus, over the years, the assumption of constantwind speed and eddy diffusivity has been relaxed to find the solution of theadvection–diffusion equation.

Efforts have been made to obtain the analytical solutions of the two-dimensional steady-state advection–diffusion equation, describing the crosswind-integrated concentrations in the atmosphere, for particular forms of windspeed and vertical eddy diffusivity (Smith 1957; Van Ulden 1978; Huang 1979;Wilson 1982; Nokes et al. 1984; Turfus 1986; Lin & Hildemann 1996; Wortmannet al. 2005; Sharan & Modani 2006). The crosswind-integrated solution is oftenused to deduce the three-dimensional concentrations of a pollutant, assuming theGaussian concentration distribution in crosswind direction, for environmentalimpact and assessment studies (Buske et al. 2007; Irwin et al. 2007). On theother hand, the crosswind-integrated concentration prohibits application to thoseproblems where the concentration plume occupies a sufficiently deep region ofthe ABL to have been influenced by the height variation of the mean winddirection. In conditions of horizontally homogeneous turbulence in the ABL, mostof the crosswind-integrated solutions (Smith 1957; Van Ulden 1978; Huang 1979;Lin & Hildemann 1996) are obtained by taking the wind speed and verticaleddy diffusivity as the power-law functions of vertical height above the ground.Turfus (1986) analysed the almost similar problem to that considered by Smith(1957) for dispersion from a line source in a reversed-flow layer by takingthe power-law wind speed and later extended the analysis for a point source.Following the approach used by Smith (1982), Nokes et al. (1984) solved thisequation for logarithmic wind speed and parabolic profile of eddy diffusivityand showed a significant improvement in the agreement between theory andobservations. Wortmann et al. (2005) used the Laplace transform technique tosolve the advection–diffusion equation for eddy diffusivity depending on both thedownwind distance (x) from the source and vertical height (z) above the groundby dividing the x-domain into subdomains. In each subdomain, the averaged valueof the eddy diffusivity and its derivative in the x variable is assumed. Recently, for

Proc. R. Soc. A (2010)


Analytical model for dispersion 385


near source dispersion, Sharan & Modani (2006) developed a model by taking thewind speed as a power-law profile and eddy diffusivity as a generalized functionof downwind distance from the source.

In all these analytical models, the wind speed is either a power law orlogarithmic profile of vertical height and similarly the eddy diffusivity has beenassumed either a power law or a parabolic profile of z or a function of downwinddistance from the source. But none of these provides a systematic approach tofind the solution with the generalized functional forms of wind speed and eddydiffusivity. Even the solution given by Nokes et al. (1984) is limited to thoseforms of eddy diffusivity and wind speed that could accurately be expandedinto the power series or polynomials and Wortmann et al. (2005) assumedconstant averaged value of eddy diffusivity and its derivative in the x variableby performing a step-wise approximation. However, these profiles of wind speedand eddy diffusivity may not simulate the ABL processes realistically becauseof its complex turbulent structure in various stability conditions. More accurateand theoretically derived representations of the wind speed and eddy diffusivitiesin the ABL have been proposed in the literature (Monin & Obukhov 1954;Nieuwstadt 1984; Holtslag & Moeng 1991; Degrazia et al. 1997; Mooney & Wilson1993; Gryning et al. 2007; and others) in various stability conditions. However,for these profiles of wind speed and eddy diffusivity in ABL, it is still not feasibleto find a generalized analytical solution of the advection–diffusion equation thatcould effectively account for the influence of turbulent structure of the ABL onthe dispersion of pollutants.

To overcome the limited efficacy of analytical dispersion models with the par-ticular forms of wind speed and vertical eddy diffusivity to deal with the turbulentdispersion of a pollutant released from a continuous source in the ABL, ageneralized analytical model describing the crosswind-integrated concentrationsis presented. An analytical scheme is described to solve the resulting two-dimensional steady-state advection–diffusion equation for wind speed as ageneralized function of vertical height above the ground and eddy diffusivityas a function of both downwind distance from the source and vertical height.This model preserves the beauty of analytical solution without compromisingthe accuracy of wind speed and eddy diffusivity to compute the crosswind-integrated concentrations.

2. Model description

(a) Advection–diffusion equation

The two-dimensional steady-state advection–diffusion equation for the crosswind-integrated concentration C (x , z) of a non-reactive contaminant released from asource in the ABL based on K -theory is given by

U (z)∂C∂x

= ∂

∂x

(Kx(x , z)

∂C∂x

)+ ∂

∂z

(Kz(x , z)

∂C∂z

), (2.1)

where U (z) is the wind speed profile in x-direction and Kx and Kz are thelongitudinal and vertical eddy diffusivities in the x- and z-directions, respectively.Equation (2.1) is derived by assuming unidirectional wind.





Under moderate to strong winds, the transport owing to longitudinal diffusionin equation (2.1) is neglected in comparison to advection and accordingly

U (z)∂C∂x

= ∂

∂z

(Kz(x , z)

∂C∂z

). (2.2)

(b) Boundary conditions

Equation (2.2) is subject to the following boundary conditions:

(i) The ground is assumed to be a perfectly reflecting surface and accordinglydiffusive flux vanishes close to the surface, i.e.

−Kz(x , z)∂C∂z

= 0 at z → zb. (2.3)

All the models in the literature assume the lower boundary at zb → 0. However,the consideration of zb → 0 as the lower boundary may be questionable as mostof the parameterizations of wind speed and eddy diffusivity are not valid in theviscous sublayer. Most of the profiles of Kz(x , z) vanish as z tends to zero, implyingthat the boundary condition (2.3) becomes an identity satisfying for all verticalconcentration gradients close to the surface.

It is evident that the most physical formulations of U and Kz , based on theMonin–Obukhov similarity theory (MOS), are applicable in the surface layer andno longer valid in the roughness sublayer. Micrometeorological measurements alsoindicate that the idealized surface layer of the similarity theory may not begineven at the top of the roughness elements of the average height ho, which isan order of magnitude larger than z0 (where z0 represents the surface roughnesslength), but at 1.5–2 times ho (Arya 1999), implying that the zb should not beless than z0. Accordingly, in the present study, we have taken zb = z0.

(ii) The pollutant is not able to penetrate through the top of theinversion/mixed layer located at height h, i.e.

−Kz(x , z)∂C∂z

= 0 at z = h. (2.4)

(iii) The pollutant is released from an elevated point source of strength Qlocated at the point (0, Hs)

U (z)C (0, z) = Qδ(z − Hs), (2.5)

where Hs is the stack height and δ is the Dirac-delta function.

(c) Solution procedure

Equation (2.2) can be re-written as

U (z)∂C∂x

= ∂Kz(x , z)∂z

∂C∂z

+ Kz(x , z)∂2C∂z2

. (2.6)

Since the boundary conditions (equations (2.3) and (2.4)) are homo-geneous, a basis of all linearly independent normalized vectors satisfying the





eigenvalue problem

d2Φ

dz2+ λ2Φ = 0 (2.7a)

dΦ

dz= 0, at z = zb, h, (2.7b)

is given by

Φn(z) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

1√(h − zb)

, n = 0,

cos(λn(z − zb))√(h − zb)/2

, n �= 0,

(2.8)

where λn = nπ/(h − zb) (n = 0, 1, 2, . . .) are the corresponding eigenvalues. Theeigenvectors Φn ’s in equation (2.8) are orthonormal. These eigenfunctions(equation (2.8)) in the form of the cosine waves arise in view of zero slop atz = zb and z = h.

The analytical solution of equation (2.2) with the boundary conditions(equations (2.3) and (2.4)) has been obtained by taking the solution of the form

C (x , z) =∞∑

n=0

An(x)Φn(z), (2.9)

where An(x), n = 0, 1, 2, . . . are the unknown coefficients of the series. To seekthese coefficients An(x), we compel the solution (2.9) to satisfy the partialdifferential equation (2.6) and accordingly, we get

∞∑n=0

dAn(x)

dxU (z) Φn(z) =

∞∑n=0

An(x)

[−Kz(x , z) λ2

n Φn(z) + ∂Kz

∂zdΦn

dz

]. (2.10)

Multiplying both the sides by eigenfunctions {Φm(z)} (m = 0, 1, 2, . . .) andintegrating it with respect to z from zb to h, we obtain a following set of first-orderordinary differential equations (ODEs)

∞∑n=0

(∫ h

zbU (z) Φm(z) Φn(z)dz

)dAn(x)

dx

=∞∑

n=0

[∫ h

zbΦm(z)

{−Kz(x , z) λ2

n Φn(z) + ∂Kz

∂zdΦn

dz

}dz

]An(x) (2.11)

This can be re-written in the matrix notationdAdx

= F A (2.12)

in which A(x) = [A0(x), A1(x), A2(x), A3(x), . . .]t (here the superscript t denotesthe transpose) is the column vector of unknown coefficients and

F = B−1 E , (2.13)





where the elements of matrices B and E are, respectively,

bmn =∫ h

zbU (z) Φm(z) Φn(z)dz (2.14a)

and emn =∫ h

zbΦm(z)

{−Kz(x , z) λ2

n Φn(z) + ∂Kz

∂zdΦn

dz

}dz . (2.14b)

In equations (2.13) and (2.14), matrices B and E appear to be rectangularmatrices for different upper integer bounds of m and n, which seems to requirethe generalized inverse of a rectangular matrix B (Greville 1959; Harwood et al.1970). Without the loss of generality, to avoid the complicated procedure involvedin finding the generalized inverse of a rectangular matrix B, we vary the m and nup to the same upper integer value N for which both B and E become the squarematrices of order N . B is a symmetric and positive definite matrix with non-zeropositive diagonal elements (electronic supplementary material 1) implying thatthe matrix B is invertible.

In equation (2.12), F is an N × N matrix of the real-valued continuousfunctions of real variable x . The system of first-order ordinary linear differentialequations (2.12) will have a unique solution (Martin 1967; Zhu 1992)

A(x) =(

exp[∫ x

0F(u) du

])A(0), (2.15)

provided the matrix F(x) satisfies the condition

F(x)

∫ x

0F(u)du −

(∫ x

0F(u)du

)F(x) =

[F(x),

∫ x

0F(u)du

]= 0, (2.16)

where the column matrix A(0) is the value of the matrix A(x) at x = 0. Thefunctional forms of Kz(x , z) used in the dispersion modelling for which the cor-responding matrix F satisfies the condition (2.16) are deduced in appendix A.

To determine A(0), solution (2.9) is substituted into the source condition(equation (2.5))

U (z)∞∑

n=0

An(0)φn(z) = Qδ(z − Hs). (2.17)

Multiplying both sides of equation (2.17) by eigenfunction Φm(z) and integratingit with respect to z from zb to h, one obtains

∞∑n=0

(∫ h

zbU (z) Φm(z) Φn(z)dz

)An(0) = QΦm(Hs) (2.18)

or in matrix notation

A(0) = B−1G, (2.19)

where G = QΦm(Hs) is a N × 1 column matrix and B is given by equation (2.14a).





(d) Computation of exponential of matrix F(x)

For all the functional forms of Kz(x , z) derived in the appendix A, the matrixF(x) in equation (2.15) can be expressed as

F(x) = f (x)M , (2.20)

where M is a square matrix with elements independent of the variablex , henceforth refers to a constant matrix. The functional form f (x) arisescorresponding to the forms of Kz as Kz(x , z) = f (x) and Kz(x , z) = f (x)g(z). Thematrix M can be transformed into the following Jordan canonical form (Horn &Johnson 1985):

M = P J P−1, (2.21)

where J is the Jordan matrix of all eigenvalues of M and P is the correspondingmatrix of all linearly independent eigenvectors. If all the eigenvalues of M aredistinct, J will be a diagonal matrix (Horn & Johnson 1985). Thus, the integrationof equation (2.20) can be written as∫ x

0F(u) du = PD P−1, (2.22)

where

D =(∫ x

0f (s)ds

)J (2.23)

is a diagonal matrix. By taking the exponential of both sides of equation (2.22),one gets

exp(∫ x

0F(u)du

)= P exp(D)P−1 (2.24)

in which the exponential of diagonal matrix D can be obtained by simply takingthe exponential of the diagonal elements of D.

Once the diagonal elements of the matrix D are known, the expression onthe left-hand side of equation (2.24) becomes known and in turn the coefficients{An(x)} will be evaluated from equation (2.15). Consequently, these coefficients’An ’s and the eigenfunctions in the series (equation (2.9)) allow to compute theconcentration C (x , z).

For a generalized F(x) in equation (2.15), satisfying the condition (2.16), itis not obvious to find the variable eigenvalues and the corresponding eigenvectormatrix for the matrix

∫x0 F(u)du. However, from the computational point of view,

we can calculate these eigenvalues and the corresponding eigenvector matrix forany preset value of x for which the coefficients of the matrix

∫x0 F(u)du become

constants. Consequently, the exponential of the matrix∫x

0 F(u)du in equation(2.15) can be calculated for any preset value of x .

We have derived the solution of equation (2.2) by assuming the lower boundaryat z = zb, avoiding the problem associated with the vanishing of Kz at the groundz = 0. For the dispersion in the ABL, zb may be taken as z0, a roughness lengthscale. The solution with the lower boundary condition at the ground z = 0 can bereadily reduced by simply taking the limit as zb → 0. The solution for a ground-level release is obtained when the effective stack height Hs tends to zero inequation (2.5). The form of the solution corresponding to a ground-level release is





similar to that of elevated release except the term Φm(Hs) tends to one in equation(2.19) and consequently the column matrix G in equation (2.19) becomes G = Q.We hasten to point out that the analogy of the approach used here is similar tothe Galerkin technique in which the basis functions are chosen satisfying theboundary conditions.

3. Comparison with existing models

In order to check the accuracy of the employed analytical solution technique givenin §2, the solution of equation (2.2) given by equation (2.9) is reduced into someparticular cases. However, it is not always feasible to reduce in the form of anexplicit expression from the solution given in §2c for all the parameterizations ofU (z) and Kz(x , z). Thus, the proposed model is compared with other analyticalmodels obtained by taking the particular forms of U (z) and Kz(x , z). As mostof the analytical models in the literature are derived by considering the lowerboundary at the ground, we have taken zb = 0 for the comparison purposes.

Most of these analytical models can be broadly classified first into two groupsbased on the wind field as (i) uniform wind speed and (ii) variable wind speed—afunction of z . In each of these groups, the model differs with the choice of theeddy diffusivity Kz(x , z).

(a) Uniform wind speed

(i) Constant eddy diffusivity Kz (x , z) = K

For constant U and K , the matrix B becomes a scalar matrix with the scalarU and E is a diagonal matrix with the elements

enn = −λ2nK . (3.1)

Thus, the matrix F in equation (2.15) will be a diagonal matrix with diagonalelements

fnn = −(K/U ) λ2n . (3.2)

In equation (2.15), the diagonalization of matrix F instantly calculates theintegration of F by just integrating the diagonal elements fnn and then takingtheir exponential leading to

An(x) = exp(

−KU

λ2nx

)An(0). (3.3)

The coefficients An(0) in equation (2.19) are given by

An(0) = QU

Φn(Hs). (3.4)

Thus, by using λn and Φn in equation (2.8) with zb → 0, the solution of equation(2.2) for constant U and Kz transforms equation (2.9) to

C (x , z) = QhU

[1 + 2

∞∑n=1

cos(λnHs) cos(λnz) exp(

−KU

λ2nx

)]. (3.5)





This is the classical solution (Seinfeld 1986) which leads to the Gaussian plumemodel for an elevated source with reflections from two parallel boundaries at z = 0and z = h.

(ii) Kz (x , z) as a function of downwind distance

For the treatment of the near source dispersion, eddy diffusivity isparameterized in terms of downwind distance x (Arya 1995; Sharan et al. 1996).In a similar manner to case §3a(i), for a uniform wind speed and Kz(x , z) = f (x),a function of downwind distance, the solution described in §2c is reduced to thefollowing form:

C (x , z) = QhU

[1 + 2

∞∑n=1

cos (λnHs) cos (λnz) exp(

−λ2n

U

∫ x

0f (x ′) dx ′

)]. (3.6)

This solution (3.6) is the same as that obtained by Sharan & Gupta (2002)for uniform wind speed and eddy diffusivity as a generalized function of x .If the eddy diffusivity in equation (3.6) is taken as constant, equation (3.6)reduces to equation (3.5). This model has been evaluated (Sharan & Gupta 2002)with the Copenhagen tracer observations by taking f (x) as a linear function ofdownwind distance.

(b) Variable wind speed as a function of vertical height above the ground

(i) Power-law wind speed and eddy diffusivity in terms of downwind distance

For predicting the crosswind-integrated concentration, wind speed and eddydiffusivity are parameterized as (Pasquill & Smith 1983)

U (z) = a zα, a = U (zr ) z−αr (3.7)

and Kz(x , z) = f (x), (3.8)

where U (zr ) is the wind speed at a reference height zr and α is the power-lawexponent that depends on the atmospheric stability and can be parameterizedby MOS theory (Huang 1979). The analytical solution of equation (2.2) withboundary conditions (2.3)–(2.5) (assuming zb = 0) with the parameterization ofwind speed (equation (3.7)) and diffusivity (equation (3.8)) is obtained by themethod of eigenfunction expansion (Sharan & Modani 2006) and is given by

C (x , z) = Qa

(1μ

− 1)

h(1−1/μ)

[1 + 1

1 − μ

(z Hs)1/2

h

]

×∞∑

n=1

J−μ

(2βnμ H 1/2μ

s

)J−μ

(2βnμz1/2μ

)J 2−μ

(2βnμ h1/2μ

) exp(

−β2n

a

∫ x

0f (x ′) dx ′

), (3.9)

where μ = 1/(α + 2), J−μ is the Bessel function of the first kind and βn ’s are theroots of

J−μ+1(2βnμ h1/2μ) = 0. (3.10)





1.2E–02(a)

(c)

(b)

(d)

9.0E–03

C/Q

(sm

–2)

6.0E–03

3.0E–03

0

1.2E–02

9.0E–03

C/Q

(sm

–2)

6.0E–03

3.0E–03

0

4.0E–03

3.0E–03

2.0E–03

1.0E–03

0

8.0E–03

6.0E–03

4.0E–03

2.0E–03

0

400 800 1200

x (m)

1600 2000 400 800 1200

x (m)

1600 2000

Figure 1. Comparisons of normalized crosswind-integrated concentration (C/Q) at ground level(at z = 0.02 m) and at plume centre line (at z = 50 m) by the present technique (viz. present)with the corresponding solutions obtained by (a) Sharan & Modani (2006) (viz. SM06), (b) Lin &Hildemann (1996) (viz. LH96), (c) Sharan & Kumar (2009) (viz. SK09) and (d) Nokes et al. (1984)(viz. Nokes) for an elevated source of stack height Hs = 50 m. (a)–(d) Circle, present: z = 0.02 m;triangle, present: z = 50 m. (a) Square, SM06: z = 0.02 m; cross, SM06: z = 50 m. (b) Square, LH96:z = 0.02 m; cross, LH96: z = 50 m. (c) Square, SK09: z = 0.02 m; cross, SK08: z = 50 m. (d) Square,Nokes: z = 0.02 m; cross, Nokes: z = 50 m.

For computational purpose, f (x) = γ U x (Sharan & Modani 2006), in whichγ = (σw/U )2 (Arya 1995), where σw refers to the standard deviation of the verticalvelocity component and U is the average wind speed.

For the comparison, the effective stack height of an elevated source is taken50 m and the mixing height is assumed to be 500 m. We have taken the parametersα = 0.29, U (zr ) = 2.1 ms−1, zr = 10 m in the power-law profile of wind speed andσw = 0.83 (ms−1), U = 5.06 (ms−1) (U is computed by taking the average ofpower-law wind profile (equation (3.7)) in ABL) in f (x).

Figure 1a compares the concentrations computed by the present technique tothat from the closed form analytical solution (3.9) for a ground-level receptor atz = 0.02 m and at plume centre line at z = 50 m with increasing x . This showsthat the concentrations computed from both the techniques overlap each otherfor all x . For z = 0.02 m, the maximum absolute error is found to be 0.5 per centoccurring near to the source. However, this error is attributed to computationalerrors in both of the solutions (2.9) and (3.9).





(ii) Both wind speed and eddy diffusivity in terms of power-law functionsof vertical height

Lin & Hildemann (1996) (after this LH96) proposed a scheme to solve equation(2.2) by using Green’s function theory for both wind speed and eddy diffusivityin terms of the power-law functions of vertical height above the ground. In theirsolution, the power-law profile for U (z) is given by equation (3.7) and eddydiffusivity is taken in the form

Kz(x , z) = g(z) = b zβ , b = Kz(zr )z−βr , (3.11)

where Kz(zr ) is the eddy diffusivity at the reference height zr . The analyticalsolution of equation (2.2) for these forms of U (z) and Kz(x , z) is (LH96)

C (x , z)/Q = α + 1ahα+1

+ α − β + 2a hα−β+2

(z Hs)(1−β)/2

×∞∑

n=1

J−μ[γn(z/h)(α−β+2)/2]J−μ[γn(Hs/h)(α−β+2)/2]J 2−μ(γn)

× exp(

−b(α − β + 2)2γ 2n

4ahα−β+2x)

(3.12)

withμ = (1 − β)/(α − β + 2), (3.13)

where γn ’s are the zeros of the following equation:

J−μ+1(γn) = 0. (3.14)

In the power-law profile of Kz(x , z) in equation (3.11), we have taken theparameters β = 0.45, Kz(zr ) = 5.0 m2 s−1 and α, U (zr ) and zr are given in the pre-vious case (§3b(i)), figure 1b compares the computed concentrations fromequation (2.9) to that with LH96 (equation (3.12)) and show that both almostoverlap each other for all x except very near to the source at z = 0.02 m, wherethe maximum absolute error is found to be less than 0.2 per cent.

(iii) Wind speed in terms of power-law functions of vertical height and Kz (x , z) isan explicit function of both downwind distance and vertical height

For near source dispersion, eddy diffusivity is also a function of downwinddistance x . Mooney & Wilson (1993) have proposed a modified functional formof eddy diffusivity in terms of both the downwind distance x and the height zabove the ground to deal with the near source dispersion. The modified form ofKz(x , z) is given in the form (Mooney & Wilson 1993)

Kz(x , z) = f (x)g(z), (3.15)

in which g(z) is any form of eddy diffusivity depending on the z and f (x) is thecorrection to g(z) for near source dispersion and is a dimensionless integrablefunction of x . A power-law parameterization of g(z), given by equation (3.11),is taken in equation (3.15). Using the power-law profile of U (z) (equation(3.7)) and the modified parameterization of Kz(x , z) (equation (3.15)), a closedform analytical solution of equation (2.2) with boundary conditions (2.3)–(2.5)





(assuming zb = 0) has been obtained by the method of separation-of-variables as(Sharan & Kumar 2009)

C (x , z)/Q = α + 1ahα+1

+ α − β + 2a hα−β +2

(z Hs)(1−β)/2

×∞∑

n=1

J−μ[γn(z/h)(α−β+2)/2] J−μ[γn(Hs/h)(α−β+2)/2]J 2−μ(γn)

× exp(

−b(α − β + 2)2 γ 2n

4 a hα−β+2

∫ x

0f (x ′)dx ′

), (3.16)

where μ is given by equation (3.13) and γn ’s are the zeros of equation (3.14).This solution is the generalization of SM06 and LH96 for near source dispersion.

By taking β = 0 and b = 1.0 in equation (3.11), the solution (3.16) is reduced toSM06 (equation (3.9)). Similarly, the solution LH96 can be obtained by takingthe function f (x) = 1.0 (as a constant function).

For comparing the solutions (equation (2.9)) and equation (3.16), f (x) is takena dimensionless function of x from Mooney & Wilson (1993). Figure 1c revealsthat the computed concentrations from the proposed technique almost exactlymatch those obtained analytically from equation (3.16) for all x except near tothe source. The maximum error is found to be less than 5 per cent (figure 1c) fornear to the source at z = 0.02 m. This error is attributed to the computationalerrors including round-off involved owing to the number of terms taken in theseries solution calculating the concentration in each model.

(iv) Wind speed and eddy diffusivity are generalized functions of vertical heightabove the ground

In the above cases, wind speed has been parameterized as a power-law profileof vertical height above the ground. However, in the atmospheric surface layer, amore realistic profile of wind speed can be obtained by using the MOS or the localsimilarity theory (Nieuwstadt 1984). Also, Kz(x , z) can have a cubic polynomialof z (O’Brien 1970) in unstable ABL and a quadratic polynomial in stable ABL(Nieuwstadt 1984), vanishing at the ground as well as at the top of the mixinglayer (Nokes et al. 1984).

In order to compare the solution (2.9) of equation (2.2) for a generalized profileof wind speed and eddy diffusivity, we have used the solution procedure proposedby Nokes et al. (1984) to the problem of turbulent flow in a long wide uniformchannel. The procedure of separation-of-variables in equation (2.2) leads to asystem of two ODEs in each of the variables x and z . The resulting second-orderODE in z is solved as a Sturm–Liouville problem using the power-series techniqueto determine the eigenvalues and associated eigenfunctions for the logarithmicprofile of wind speed and parabolic eddy diffusivity. A slightly changed profile ofU (z) from Nokes et al. (1984) to avoid the negative values of wind speed nearto the surface is taken. Accordingly, we have assumed the following forms oflogarithmic wind speed and parabolic eddy diffusivity:

U (z) = U [1 + ρ(1 + ln([z + z0m]/h))] (3.17)

and Kz(x , z) = g(z) = (η/ρh2)(h − z)(z + z0m), (3.18)





where U is the depth-averaged velocity, ρ = u∗/kU is the dimensionlessparameter, η = u2∗h/U is the diffusivity coefficient and z0m is chosen such thatU (z) becomes zero at z = 0.

For comparison, we have taken the parameter’s values in profiles (3.17) and(3.18) from Nokes et al. (1984). Figure 1d demonstrates that the results computedusing the technique described in this paper are in good agreement to thoseobtained from the one based on the power-series method adopted by Nokes etal. (1984) for Hs = 50 m and h = 500 m at z = 0.02 and z = 50 m. The maximumabsolute error for ground-level receptor is found to be 2 per cent.

4. Results and discussion

The model given here is general in the sense that it can use any form ofparameterization of wind speed U (z) and eddy diffusivity Kz(x , z) as an explicitfunction of variables x and z . The comparison results demonstrate that thecrosswind-integrated concentrations computed by the proposed technique arein very good agreement with those obtained from the analytical solutionscorresponding to the particular functional forms of U (z) and Kz(x , z). Thecomputed concentrations from the proposed technique almost exactly match forall x except near to the source.

An analysis for the stability of the solution (2.9) is given in the electronicsupplementary material 2. The mathematical analysis shows that the presentsolution is asymptotically stable. Also, based on the computations (electronicsupplementary material 3), it is observed that N = 250 terms are adequate toobtain a converged solution (equation (2.9)).

(a) Model evaluation

The performance of the present model has been analysed from the Copenhagendiffusion experiment in unstable conditions and the Hanford and PrairieGrass experiments in stable conditions. We have used the standard statisticalperformance measures (i) normalized mean square error (NMSE), (ii) fractionalbias (FB), (iii) correlation coefficient (COR), (iv) fractional variance (FS), and (v)factor of two (FAC2) that characterize the agreement between model predictionand observations (Sharan & Kumar 2009). A perfect model would have theidealized values as NMSE = FB = FS = 0 and COR = FAC2 = 1.

(i) Copenhagen diffusion experiment in convective conditions

A tracer experiment in unstable and neutral conditions was conducted in thenorth Copenhagen area (Gryning & Lyck 1984; Gryning et al. 1987). In thisexperiment, a tracer SF6 was released without buoyancy at a tower height of 115 mand samples were collected on different arcs at 2 m above the ground surface. Adetail description of meteorological and tracer data is provided by Gryning &Lyck (1984) and Gryning et al. (1987). The values of parameters such as h, frictionvelocity (u∗), Obukhov length (L), σv, σw and U observed during the experimentare taken from Gryning et al. (1987). The site was primarily residential with aroughness length of 0.6 m and thus the terrain was considered as an urban terrainin the computation.





Parameterization of U(z) in unstable conditionNormally, U (z) is parameterized in the dispersion model using an empirical

formulation in terms of power-law function of z . The wind speed profile U (z) isalso parameterized based on surface layer theory and Monin–Obukhov scaling.However, the profile based on the surface layer theory is valid only within thesurface layer. This profile has been extended in the entire ABL by assuming aconstant profile of wind speed above the surface layer. Recently, Gryning et al.(2007) have proposed a formulation for the wind speed profile in the entire ABLby extending the surface layer profile of wind speed over homogeneous terrainbeyond the surface layer. In this profile, the local length scale in wind speed iscomposed of three component length scales. In the surface layer, the first lengthscale (LSL) is taken to increase linearly with height with a stability correctionfollowing MOS theory. Above the surface layer, the second length scale (LMBL)becomes independent of height but not stability, and at the top of the boundarylayer, the third length scale is assumed to be negligible.

For unstable atmospheric conditions, the wind speed profile is given as(Gryning et al. 2007)

U (z) = u∗k

[ln

(zz0

)− ψ

( zL

)+ z

LMBL− z

h

(z

2LMBL

)], (4.1)

where

ψ( zL

)= 3

2ln

(1 + ξ + ξ 2

3

)− √

3 tan−1(

1 + 2ξ√3

)+ π√

3(4.2)

in which ξ = (1 − 12 z/L)1/3 and LMBL is given by

LMBL = h2

⎡⎣{(

ln(

u∗fc z0

)− B(μ)

)2

+ A2(μ)

}1/2

− ln(

hz0

)+ ψ

(hL

)⎤⎦

−1

, (4.3)

where fc is the Coriolis parameter, k=0.4 is von Karman constant and A andB are the resistance law functions depending on the dimensionless stabilityparameter μ = u∗/fcL in the steady-state ABL, or h/L in the evolving ABL.For the neutral atmosphere, the values of A ≈ 4.9 and B ≈ 1.9 are rather wellestablished. However, in stable and unstable conditions, the stability dependenceof these functions A and B is a matter of discussion (Zilitinkevich & Esau 2005).Owing to the ambiguity in the formulation of the functions A and B, Gryninget al. (2007) have also proposed an empirical formula for the length scale LMBLin both unstable and stable conditions as

LMBL = u∗fc

[−2 ln

(u∗

fc z0

)+ 55

]−1

exp(

(u∗/fc z0)2

400

). (4.4)

In the present study, LMBL is taken from equation (4.4) to compute the windprofile from equation (4.1).

Parameterization of Kz(x , z) in unstable conditionsIn the dispersion models, Kz is parameterized as a function of the height above

the ground, z , only. For the near source dispersion, Kz has also been parameterizedas a function of downwind distance x (Arya 1995; Sharan & Modani 2006).





10(a) (b)

1

0.10 2 4 6 8 10 12

12

10

8

6

4

2

0

Cobs/Q × 10–4 (sm–2)

Cpr

ed/Q

× 1

0–4 (s

m–2

)

(Cpr

ed/Q

)/(C

obs/

Q)

2 4 6 8 10 12

Cobs/Q × 10–4 (sm–2)

Figure 2. Comparisons of predicted (Cpred/Q) and observed concentration (Cobs/Q) for theCopenhagen diffusion experiment in unstable conditions for two parameterizations of Kz (x , z)as (i) a function of z only (i.e. Kz (x , z) = Kz (z)) and (ii) an explicit function of x and z (i.e.Kz (x , z) = Kz (z) f (x)) by (a) the ratio of Cpred/Q and Cobs/Q with Cobs/Q and (b) the Q–Qplot between Cobs/Q and Cpred/Q. Middle lines in both plots are one-to-one line and outer linesare the lines with a factor of two. (a,b) Triangle, Kz (z); square, Kz (z)∗f (x).

Mooney & Wilson (1993) have used a modified form of Kz as a function of bothvariables x and z as

Kz(x , z) = K ′z(z) [1 − exp(−x/L1)] , (4.5)

where K ′z is the functional form of eddy diffusivity depending on z only and L1 =

U (Hs)τ (Hs) is the along-wind length scale across which the diffusivity achievesits ‘far-field’ value and τ is the Lagrangian time scale at the source height.

The following form of K ′z(z) for unstable conditions as a function of z derived

by Degrazia et al. (1997) in terms of the convective scaling parameters is used inthe present study in formulation (4.5)

K ′z(z) = 0.22(w∗h)

[ zh

(1 − z

h

)]1/3[1 − exp

(−4z

h

)− 0.0003 exp

(8zh

)], (4.6)

where w∗ is the convective velocity scale and computed from u∗ using the relationw∗ = u∗(−h/(kL))1/3 (Seinfeld 1986). The Lagrangian time scale τ at the sourceheight z = Hs is parameterized as (Mooney & Wilson 1993)

τ(z) = K ′z(z)/σ

2w , (4.7)

where σw is the vertical turbulent intensity.Notice that in the model, effectively, Kz(x , z) has been parameterized as

a function of both x and z . However, in order to analyse the contributionarising from the dependency of Kz(x , z) on x in equation (4.5), we have alsoparameterized Kz(x , z) = K ′

z(z) as a function of z only by equation (4.6). Figure 2ashows that the normalized crosswind-integrated concentrations computed fromthe present model in unstable conditions are in good agreement with theobservations. All the points are predicted within a factor of two. In addition,it reveals that the predicted concentrations for both the parameterizations of





Table 1. Statistical performance measures for normalized crosswind-integrated concentrations formodel results with the Copenhagen diffusion experiment (unstable conditions), Hanford diffusionexperiment (in stable conditions) and stable runs of the Prairie Grass experiment.

experiments models Kz (x , z) NMSE COR FB FS FA2

Copenhagen present Kz (z) 0.086 0.90 0.12 0.65 1.0(unstable) Kz (z )f (x) 0.075 0.88 0.039 0.53 1.0

GILTT Kz (z) 0.06 0.92 −0.14 −0.02 1.0(present) (0.06) (0.91) (0.06) (0.46) (1.0)

Hanford (stable) present Kz (z) 0.20 0.91 −0.13 0.15 0.73Kz (z)f (x) 0.21 0.91 −0.14 0.12 0.73

SM06 f (x) 0.52 0.87 0.32 0.39 0.53Prairie Grass (only present Kz (z) 0.06 0.972 −0.16 0.12 0.95

stable runs) Kz (z)f (x) 0.06 0.973 −0.16 0.11 0.95

Kz—depending on either both x and z or z only—are comparable to each other.In fact, the predicted concentrations from Kz(x , z) appear to be relatively closerto the observations than those obtained with Kz as a function of z only. It isnoticed that the highest concentrations correspond to the shortest downstreamdistances, where the x-dependency of the Kz is expected to be significant. Theextent of the dependency of the parameterization of Kz on x can be analysed bythe contribution of the term e−x/L1 (equation (4.5)). When Hs → 0 correspondingto a ground-level release, it leads to L1 → 0 and the contribution of the terme−x/L1 becomes negligible. It implies that Kz(x , z) reduces to K ′

z(z) and Kz(x , z)will be a function of z only. For Hs �= 0, the value of L1 increases as Hs increasesand thus the contribution of the term e−x/L1 will appear in the parameterizationof Kz(x , z) in terms of both x and z for an elevated source. For Hs = 115 m inthis experiment, the difference between the results for the parameterizations ofKz(x , z) and K ′

z(z) is clearly seen (figure 2 and table 1) and it shows that thedependency of x in the Kz gives a relatively better result. The unpaired analysis,Q–Q plot (figure 2b) drawn by arranging both the observed and the predictedconcentrations in increasing order of their magnitudes (Venkataram 1999), showsthat the predicted concentrations are close to the one-to-one line observations forboth parameterizations of Kz(x , z). This fact is re-affirmed from the statisticalmeasures given in table 1. The overall statistical measures indicate that thepresent model has a good correlation with the observations.

We have also compared the results from the present model with amodel obtained by the generalized integral transform technique (GILTT)given by Wortmann et al. (2005). For this purpose, we have taken thesame parameterizations of the wind speed and eddy diffusivity as used byWortmann et al. (2005). Statistical indices in table 1 indicate that the resultsfrom the present model are comparable with those obtained using GILTT.

(ii) Hanford tracer experiment in stable conditions

In stable conditions, a diffusion experiment was conducted in May–June,1983, at the Hanford site in eastern Washington (46◦34 N, 119◦36 W) USA(Doran & Horst 1985). The terrain at the Hanford Washington site is flat with





a displacement height of approximately 1.4 m (Doran & Horst 1985). In the SF6tracer experiment, six test runs were conducted in 1983 and the concentrations ofSF6 were measured at five sampling arcs 100, 200, 800, 1600 and 3200 m downwindfrom a fixed release point. The tracer was released at 2 m and was sampled at1.5 m height above the ground. Values of the meteorological parameters h, u∗ andL in the evaluation of the model are taken from Doran et al. (1984). Values ofsurface roughness length z0 for each run are adopted from the re-analysis of thedata by Kim & Larson (2001).

Parameterization of U(z) in stable conditionsFor stable conditions, the extended profile of wind speed into entire ABL is

given as (Gryning et al. 2007)

U (z) = u∗k

[ln

(zz0

)+ bz

L

(1 − z

2h

)+ z

LMBL− z

h

(z

2LMBL

)], (4.8)

where b = 4.7 and LMBL in stable conditions is

LMBL = h2

⎡⎣{(

ln(

u∗fc z0

)− B(μ)

)2

+ A2(μ)

}1/2

− ln(

hz0

)− bh

2L

⎤⎦

−1

. (4.9)

As mentioned earlier, owing to the uncertainty involved in the determination ofresistance law functions A and B as a function of μ in stable conditions, we haveadopted the empirical relation (4.4) for LMBL to calculate the wind profile fromequation (4.8) in the present study.

Parameterization of Kz(x , z) in stable conditionsFor dispersion in stable conditions, Kz(x , z) is parameterized by the same

expression given in equation (4.5). However, the following form of K ′z(z), derived

by Degrazia & Moraes (1992) from the local similarity theory and the statisticaldiffusion theory, is taken:

K ′z(z) = u∗h

[0.33(1 − z/h)α1/2(z/h)

][1 + 3.7 (z/h) (h/Λ)]−1 , (4.10)

where Λ is the local Obukhov length, defined by Λ = L(1 − z/h)(3α1/2)−α2 , in whichα1 and α2 are constant parameters. We have used α1 = 3/2 and α2 = 1, derived byNieuwstadt (1984) from the second-order closure model for stationary turbulencewith constant cooling rate. In stable conditions, σw in equation (4.7) is computedby the following expression given by Nieuwstadt (1984):

σw = 1.4u∗(1 − z/h)0.75. (4.11)

In order to analyse the contribution arising from the dependency of Kz(x , z) onx in equation (4.5), we have also parameterized Kz(x , z) = K ′

z(z) as a functionof z only by equation (4.10). Figure 3a shows that the normalized crosswind-integrated concentrations computed from the present model in stable conditionsare in good agreement with the observations and it predicts 73 per cent casesin a factor of two to observations. In addition, it reveals that the predictedconcentrations for both the parameterizations of Kz -depending on either both xand z or z only—are comparable to each other. This is justifiable as both the





10(a)

1

0.1

(Cpr

ed/Q

)/(C

obs/

Q)

(b) 10

8

6

4

2

0

Cpr

ed/Q

× 1

0–2 (s

m–2

)

0 2 4 6 8 10

Cobs/Q × 10–2 (sm–2)2 4 6 8 10

Cobs/Q × 10–2 (sm–2)

Figure 3. Comparisons of predicted (Cpred/Q) and observed concentration (Cobs/Q) for Hanfordtracer experiment in stable conditions for two parameterizations of Kz (x , z) as (i) a function of zonly (i.e. Kz (x , z) = Kz (z)) and (ii) an explicit function of x and z (i.e. Kz (x , z) = Kz (z) f (x)) by(a) the ratio of Cpred/Q and Cobs/Q with Cobs/Q and (b) the Q–Q plot between Cobs/Q andCpred/Q. Middle lines in both plots are one-to-one line and outer lines are the lines with a factorof two. (a,b) Triangle, Kz (z); square, Kz (z)∗f (x).

parameterizations become identical because of the negligible contribution of theterm e−x/L1 in equation (4.5) for ground-level releases. The unpaired analysis,Q–Q plot in figure 3b shows that the predicted concentrations are close to theone-to-one line observations for both of the parameterizations of Kz(x , z). This isre-affirmed from the values of statistical measures given in table 1.

We have also compared the results from the present model with a model fornear source dispersion proposed by Sharan & Modani (2006) (SM06). The overallstatistical measures in table 1 indicate that the present model predicts reasonablybetter than SM06. The NMSE of the present model is 0.20 whereas for SM06 itis 0.52. The COR shows that the present model has a good correlation with theobservations. It is worth noting that the model leads to negative FB, i.e. a generaloverestimation of the concentration data is pointed out.

(iii) Prairie Grass experiment

The Prairie Grass experiment (Barad 1958) was conducted in O’Neill,Nebraska, during the summer of 1956 and is a standard and widely used dataset ofdispersion observations used for the evaluation of dispersion models for ground-level releases of continuous plume over flat terrain. In all 68 test runs of thisexperiment, a tracer sulphur dioxide (SO2) was released without buoyancy ata height of 0.46 m (except four runs# 65–68 in which the release height was1.5 m) and the samples of the concentrations of SO2 were measured at 1.5 mabove the ground surface on five sampling arcs 50, 100, 200, 400 and 800 mdownwind from the source. We have used the digital version of the dataset(http://www.dmu.dk/International/Air/Models/Background/ExcelPrairie.htm)for normalized crosswind-integrated concentrations and meteorological variables.In this dataset, the meteorological parameters L, u∗, w∗ and h are given and thesurface roughness length z0 was reported as 0.006 m.


http://www.dmu.dk/International/Air/Models/Background/ExcelPrairie.htm




10(a) (b)

1

0.1

(Cpr

ed/Q

)/(C

obs/

Q)

0 300 600 900 1200 1500

Cobs/Q × 10–4 (sm–2)0 300

300

600

600

900

900

1200

1200

1500

1500

Cobs/Q × 10–4 (sm–2)

Cpr

ed/Q

× 1

0–4 (s

m–2

)

Figure 4. Comparisons of predicted (Cpred/Q) and observed concentration (Cobs/Q) for stableruns of the Prairie Grass experiment for two parameterizations of Kz (x , z) as (i) a function of zonly (i.e. Kz (x , z) = Kz (z)) and (ii) an explicit function of x and z (i.e. Kz (x , z) = Kz (z) f (x)) by(a) the ratio of Cpred/Q and Cobs/Q with Cobs/Q and (b) the Q–Q plot between Cobs/Q andCpred/Q. Middle lines in both plots are one-to-one line and outer lines are the lines with a factorof two. (a,b) Triangle, Kz (z); square, Kz (z)∗f (x).

In the evaluation, the computations are carried out with this dataset containingthe runs in all stability conditions from very unstable to neutral and verystable. However, we report the results for the runs in stable conditions only.In this stability, we have also abandoned the runs of very stable conditions(h/L > 10) in which the turbulence becomes very weak, sporadic, intermittentand no longer continuous in time and space (Holtslag & Nieuwstadt 1986).In computations, the release and sampling heights are taken as 0.46 and1.5 m, respectively. We have used the extended wind profile (equation (4.8))into the entire ABL and the vertical eddy diffusivity profile is taken as thesame as that used in the evaluation of the Hanford diffusion experiment instable conditions. The values of σw in equation (4.7) are computed from theexpression (4.11).

The scatter diagram between the ratios of predicted to observed crosswind-integrated concentrations and the observed concentrations is shown in figure 4a.In order to analyse the contribution arising from the dependency of Kz(x , z) onx in equation (4.5), we have also parameterized Kz(x , z) = K ′

z(z) as a function ofz only (equation (4.10)). Figure 4a shows that the predicted concentrations fromthe present model are in good agreement with the observations. The predictedconcentrations for both the parameterizations of Kz—depending on either bothx and z or z only—are comparable to each other. This is expected due tothe negligible contribution of the term e−x/L1 in equation (4.5) for a ground-level release. For both of the parameterizations of Kz(x , z), the model predicts95 per cent cases in the factor of two to observations and the NMSE and FBbetween the computed and the observed concentrations are, respectively, 0.06and −0.16 (table 1). The Q–Q plot (figure 4b) reconfirms that the concentrationspredicted from the present model are close to the one-to-one line for bothparameterizations of Kz(x , z).





The results discussed here in both unstable and stable conditions are forU (z) based on the empirical parameterization of LMBL in equation (4.4) whichwas obtained from different datasets and is observed better for stable thanunstable conditions (Gryning et al. 2007). Owing to the inadequate meteorologicaldata in the Copenhagen, Hanford and Prairie Grass experiments, we couldnot find the appropriate resistance law functions A and B to compute LMBLrequired in the computation of U (z). However, in unstable conditions for theCopenhagen diffusion experiment, we tried to use the empirical relations offunctions A and B proposed by Arya (1975) based on the data from the twosites (O’Neill in Nebraska, USA, and Hay in New South Wales, Australia).For Kz(x , z) = K ′

z(z) in unstable conditions, the improved parameterizationof LMBL in the wind profile based on these resistance law functions revealsthat the NMSE and FB reduce from 0.086 to 0.05 and 0.12 to 0.036,respectively. It also gives a better correlation (COR = 0.92) with the observationsin comparison to the COR = 0.90 for empirical relation of LMBL (equation(4.4)). It shows that a realistic parameterization of wind profile can lead toa significant improvement in the predication of concentrations of pollutantsin the ABL.

Here, in comparison with measurements, it is noted that the model tends tounderestimate (FB > 0) for unstable conditions, but to overestimate (FB < 0)for stable conditions. The underpredicting (overpredicting) trends of theconcentrations from the model in unstable (stable) conditions may be attributedto a slight overestimation (underestimation) of the computed mean wind speedsfrom the formulation of Gryning et al. (2007) in comparison to those observed.

Here, the model is evaluated in stable conditions with the observationscorresponding to the releases from a source near to the ground. On the otherhand, the model is proposed here for both elevated and ground-level sources.Thus, the model needs to be evaluated further once the diffusion data becomeavailable for an elevated source in stable conditions.

(b) Three-dimensional concentration at a specific location in the atmosphere

The concentrations obtained from solution (2.9) are two-dimensional andthe crosswind dispersion term in advection–diffusion equation (2.2) is notexplicitly treated. However, if one is interested in the impact or exposurestudies of the concentrations at a given location (x , y, z) in the atmosphererather than the concentration integrated along the crosswind direction y, thelateral diffusion term needs to be included to calculate the three-dimensionalconcentration c(x , y, z). Very limited analytical solutions are available by takingthe lateral diffusion term in the advection–diffusion equation even for the specificforms of wind speed and eddy diffusivities (Yeh & Huang 1975; Huang 1979).However, based on the analysis of Prairie Grass and some other historical tracerexperiments of atmospheric dispersion, Irwin et al. (2007) concluded that theground-level crosswind concentration profile of a dispersing plume on average iswell characterized as having a Gaussian shape, which is also predicted by allatmospheric transport and diffusion models, regardless of their sophistication.Thus, by assuming the Gaussian concentration distribution in crosswind direction(Huang 1979; Buske et al. 2007; Irwin et al. 2007), the steady-state concentrationof a pollutant released from a point source at (0, 0, Hs) in a three-dimensional





domain can be described as

c(x , y, z) = C (x , z)exp(−y2/2σ 2

y )√2πσy

, (4.12)

where C (x , z) is the crosswind-integrated concentration and can be obtainedfrom equation (2.9) for any generalized profiles of U (z) and Kz(x , z) and σy isthe standard deviation of the concentration distribution in crosswind direction(Huang 1979; Pasquill & Smith 1983).

(c) Issues and limitations

So far, an analytical dispersion model describing the crosswind-integratedconcentrations is presented and evaluated with the observations and itsapplication to the three-dimensional dimension is also described. The presentwork is subject to the underlying assumptions and limitations (electronicsupplementary material 4) associated with (i) models based on K -theory,(ii) solution with explicit functional forms of Kz(x , z), (iii) neglect oflongitudinal diffusion, (iv) applicability in low wind speed conditions, (iv) theparameterization of Kz(x , z) used in the evaluation and (v) neglecting the veeringin wind speed with vertical height.

5. Conclusions

For the turbulent dispersion of a pollutant released from a continuoussource in the ABL, a generalized analytical model describing the crosswind-integrated concentrations is presented. An analytical scheme is described tosolve the resulting two-dimensional steady-state advection–diffusion equation forhorizontal wind speed as a generalized function of vertical height above the groundand eddy diffusivity as a function of both downwind distance from the sourceand vertical height. On the basis of a mathematical analysis, the appropriatefunctional forms of vertical eddy diffusivity are derived. Particular cases of thismodel are deduced for (i) uniform wind speed and eddy diffusivity leading to theclassical Gaussian plume model and (ii) uniform wind speed and eddy diffusivityas a function of the downwind distance. By taking the particular forms of windspeed and vertical eddy diffusivity, the solutions of the resulting advection–diffusion equation obtained from the present technique are found to be close tothose derived using classical techniques. Some of the limitations associated withthis work are pointed out.

The model is evaluated with the observations obtained from the Copenhagendiffusion experiment in unstable conditions and the Hanford and Prairie Grassexperiments in stable conditions. In evaluating the model, we have used the rec-ently proposed formulation (Gryning et al. 2007) for the wind speed in theentire ABL. From the analysis of statistical measures for diffusion experimentsin different stability conditions, it is concluded that the present model performswell with the observations.

The present model can also be used for the concentration distribution of apollutant released from a line source perpendicular to the direction of the meanwind. The model is also useful to study the turbulent dispersion from a steadytwo-dimensional horizontal source in a wide, open channel for a generalized profileof wind flow and eddy diffusivity as discussed by Nokes et al. (1984).





Appendix A. Evaluation of the forms of Kz(x , z)

The commutative condition (2.16) may not be satisfied for all generalized formsof F(x) corresponding to Kz(x , z). To find some of these suitable forms of Kz(x , z),we have used the following theorem:

Theorem (Martin 1967). Let F(x) be the continuous matrix on an interval X.Then the following are equivalent for all x0, x1 and x2 in X:

F(x1)F(x2) = F(x2)F(x1) (A 1)∫ x1

x0

F(s)ds∫ x2

x0

F(u)du =∫ x2

x0

F(u)du∫ x1

x0

F(s)ds (A 2)

and F(x1)

∫ x2

x0

F(u)du =∫ x2

x0

F(u)duF(x1). (A 3)

Notice that equation (A 3) of the theorem implies that the condition (2.16)is satisfied in a natural way. According to the theorem, it is adequate to verifyone of these parts to ensure that the condition (2.16) is satisfied for the matrixF(x). Based on this theorem, we find the following functional forms of Kz(x , z)for matrix F satisfying the condition (2.16):

Kz(x , z) = constant, (A 4)

Kz(x , z) = g(z), (A 5)

Kz(x , z) = f (x) (A 6)

and Kz(x , z) = f (x) g(z), (A 7)

where f (x) and g(z) are the generalized functions of x and z , respectively. In spiteof these forms of Kz(x , z), there may be other implicit forms of Kz for matrix Fsatisfying the commutative condition (2.16). Within the analytical framework,a choice of an implicit functional form of x and z for the eddy diffusivity Kz isnot obvious. However, numerical integration will be needed with an appropriateimplicit functional form of Kz even for evaluating the matrix F(x). If such an F(x)satisfies condition (2.16), the solution described in §2c can be used for computingthe concentration distribution. Most of the parameterizations for Kz(x , z) usedin atmospheric dispersion modelling are in the forms defined in equations(A 4)–(A 7) satisfying condition (2.16). Thus, in the present manuscript, we haveused these commonly used parameterizations of Kz to find the solution, in theanalytical framework, of the resulting partial differential equation (equation (2.2))describing the crosswind-integrated concentration.

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